Page 1 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Controllability of space-time fractional diffusive
and super diffusive equations
Mahamadi Jacob WARMA
George Mason University, Fairfax, Virginia (USA)
The research of the author is partially supported by
US Air Force Office of Scientific Research (AFOSR)
US Army Research Office (ARO)
AIMS-Cameroon Research Center: October 13, 2020
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 2 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
1 Objectives of the talk
2 Space-time fractional order operators
3 Controllability results for space-time fractional PDEs
4 The case of the fractional heat equation
5 Open problems
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 3 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Outline
1 Objectives of the talk
2 Space-time fractional order operators
3 Controllability results for space-time fractional PDEs
4 The case of the fractional heat equation
5 Open problems
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 4 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
The considered problem
In this talk we consider the following system of evolution equation:
∂
α
t u(t, x) + (−∆)su(t, x) = f χω in Ω × (0,T),
+ Intial conditions,
+ Boundary conditions.
(1.1)
Here α > 0 is a real number, 0 < s ≤ 1, Ω ⊂ R
N is a bounded open set
with Lipschitz continuous boundary ∂Ω, (−∆)s
is the fractional
Laplacian and ∂
α
t
is a fractional time derivative of Caputo type. These
notions will be defined later.
If α = 1 (resp. α = 2) we have the heat (resp. wave) equation.
If 0 < α < 1 such an equation is said to be of slow diffusion.
If 1 < α < 2, then it is said to be of super diffusion.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 5 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Questions
How to define the fractional Laplace operator (−∆)s
?
How to define a time fractional derivative ∂
α
t
?
Which initial and boundary conditions make the system (1.1) well
posed as a Cauchy problem?
Is there a control function f localized in a nonempty open set
ω ⊂ Ω such that solutions of the system can rest at some time
T > 0? In other words, is such system null controllable?
Given a target, is there a function f localized in a nonempty open
set ω ⊂ Ω such that solutions of the system reach the given target
at time T > 0? In other words, is such system exactly controllable?
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 6 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Outline
1 Objectives of the talk
2 Space-time fractional order operators
3 Controllability results for space-time fractional PDEs
4 The case of the fractional heat equation
5 Open problems
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 7 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
The fractional Laplacian: Using Fourier Analysis
Using Fourier analysis, we have that the fractional Laplace operator
(−∆)s
can be defined as the pseudo-differential operator with symbol
|ξ|
2s
. That is,
(−∆)s
u = CN,sF
−1
|ξ|
2sF(u)
,
where F and F
−1 denote the Fourier, and inverse Fourier, transform,
respectively, and C(N,s) is an appropriate normalizing constant
depending only on N and s.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 8 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
The fractional Laplacian: Using Singular Integrals
Let 0 < s < 1 and ε > 0 be real numbers. For a measurable function
u : R
N → R we let
(−∆)s
εu(x) = CN,s
Z
{y∈RN : |x−y|>ε}
u(x) − u(y)
|x − y|
N+2s
dy, x ∈ R
N
.
The fractional Laplacian (−∆)s
is defined for x ∈ R
N by the following
singular integral:
(−∆)s
u(x) = CN,sP.V. Z
RN
u(x) − u(y)
|x − y|
N+2s
dy = lim
ε↓0
(−∆)s
εu(x),
provided that the limit exists, where CN,s
:=
s2
2sΓ
N+2s
2
π
N
2 Γ(1 − s)
. Here, Γ
denotes the usual Euler-Gamma function.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 9 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
The fractional Laplacian: Using the Caffarelli-Silvestre extension (CPDE,
2007)
Let 0 < s < 1. For u : R
N → R in an appropriate space, consider the
harmonic extension W : [0, ∞) × R
N → R. That is the unique weak
solution of the Dirichlet problem
(
Wtt +
1−2s
t Wt + ∆xW = 0 in (0, ∞) × R
N ,
W (0, ·) = u in R
N .
(2.1)
Then the fractional Laplace operator can be defined as
(−∆)s
u(x) = −ds
lim
t→0+
t
1−2sWt(t, x), x ∈ R
N
,
where the constant ds
is given by ds
:= 22s−1 Γ(s)
Γ(1 − s)
. This is known in
the literature as the Caffarelli-Silvestre extension.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 10 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
All the definitions coincide
Let 0 < s < 1. Then
(−∆)s
u(x) =CN,sF
−1
|ξ|
2sF(u)
=CN,sP.V. Z
RN
u(x) − u(y)
|x − y|
N+2s
dy
= − ds
lim
t→0+
t
1−2sWt(t, x),
where we recall that W : [0, ∞) × R
N → R is the harmonic
extension which is the unique weak solution of the Dirichlet problem
(2.1).
It is clear that (−∆)s
is a nonlocal operator. That is,
supp[(−∆)s
u] 6⊂ supp[u].
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 11 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Derivation of singular integrals: Long jump random walks
Let K : R
N → [0, ∞) be an even function such that
X
k∈ZN
K(k) = 1. (2.2)
Given a small h > 0, we consider a random walk on the lattice hZ
N .
We suppose that at any unit time τ (which may depend on h) a
particle jumps from any point of hZ
N to any other point.
The probability for which a particle jumps from a point hk ∈ hZ
N
to the point hk
̃ is taken to be K(k − k
̃) = K(k
̃ − k). Note that,
differently from the standard random walk, in this process the
particle may experience arbitrarily long jumps with small probability.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 12 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Long jump random walks: Continue
Let u(x,t) be the probability that our particle lies at x ∈ hZ
N at
time t ∈ τZ.
Then u(x,t + τ ) is the sum of all the probabilities of the possible
positions x + hk at time t weighted by the probability of jumping
from x + hk to x. That is,
u(x,t + τ ) = X
k∈ZN
K(k)u(x + hk,t).
Using (2.2) we get the following evolution law:
u(x,t + τ ) − u (x,t) = X
k∈ZN
K(k) [u(x + hk,t) − u(x,t)] . (2.3)
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 15 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Long jump random walks: Conclusion
We have shown above that a simple random walk with possibly long
jumps produces, at the limit, a singular integral with a homogeneous
kernel.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 16 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
The limit as s ↑ 1
Let u, v be smooth functions with compact support in Ω. That is,
u, v ∈ D(Ω). Then the following holds.
lim
s↑1−
Z
RN
v(−∆)s
udx = −
Z
Ω
v∆udx =
Z
Ω
∇u · ∇v dx.
Proof
Using a result due to Bourgain, Brezis and Mironescu we get:
lim
s↑1−
Z
RN
u(−∆)s
udx
= lim
s↑1
s2
2s−1Γ
N+2s
2
π
N
2 (1 − s)Γ(1 − s)
(1 − s)
Z
RN
Z
RN
|u(x) − u(y)|
2
|x − y|
N+2s
dxdy
=
Z
RN
|∇u|
2
dx =
Z
Ω
|∇u|
2
dx = −
Z
Ω
u∆udx.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 17 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Question
1 What are the Dirichlet and Neumann Boundary Conditions for the
fractional Laplace operator (−∆)s
?
2 To obtain an explicit and a rigorous answer to the above question,
we need the following notions.
We need some appropriate Sobolev spaces.
We need a notion of a (fractional) normal derivative.
We also need an integration by parts formula for (−∆)s
. That
is, an appropriate Green type formula for (−∆)s
.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 19 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Fractional order Sobolev Spaces: Continue
Let D(Ω) be the space of test functions on Ω. Let
W
s,2
0
(Ω) = D(Ω)Ws,2
(Ω)
,
and
W
s,2
0
(Ω) = n
u ∈ W s,2
(R
N
) : u = 0 a.e. on R
N
\ Ω
o
.
1 There is no obvious inclusion between W
s,2
0
(Ω) and W
s,2
0
(Ω).
2 If Ω ⊂ R
N is Lipschitz, then we have the following situation.
If s 6=
1
2
, then W
s,2
0
(Ω) = W
s,2
0
(Ω).
If s =
1
2
, then W
s,2
0
(Ω) is a proper subspace of W
s,2
0
(Ω).
For arbitrary bounded open sets, the relation can be found in
W. Potential Analysis, 2015.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 20 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
The Dirichlet problem for (−∆)s
Let g ∈ C(∂Ω). The classical Dirichlet problem for ∆ is given by
∆u = 0 in Ω, u = g on ∂Ω.
Let g ∈ C(∂Ω). Then the Dirichlet problem
(−∆)s
u = 0 in Ω, u = g on ∂Ω, (2.6)
is not well-posed. This follows from the fact that
(−∆)s
u(x) = CN,s
Z
Ω
u(x) − u(y)
|x − y|
N+2s
dy + CN,s
Z
RN \Ω
u(x) − u(y)
|x − y|
N+2s
dy.
Let g ∈ C0(R
N \ Ω). The well-posed Dirichlet problem is given by
(−∆)s
u = 0 in Ω, u = g in R
N
\ Ω.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 21 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
The zero Dirichlet exterior condition (EC) for (−∆)s
1 The zero Dirichlet BC for ∆ is given by u = 0 on ∂Ω.
2 The zero Dirichlet EC is characterized by u = 0 in R
N \ Ω.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 22 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
How to define a ”fractional” normal derivative?
Recall that if u is a smooth function defined on a smooth open set
Ω, then the normal derivative of u is given by
∂u
∂ν := ∇u · ~ν,
where ~ν is the normal vector at the boundary ∂Ω.
For 0 < s < 1 and a function u defined on R
N we let
Nsu(x) = CN,s
Z
Ω
u(x) − u(y)
|x − y|
N+2s
dy, x ∈ R
N
\ Ω,
provided that the integral exists. This is clearly a nonlocal operator.
Ns
is well-defined and continuous from W s,2
(R
N ) into L
2
(R
N \ Ω).
We call Nsu the nonlocal normal derivative of u.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 23 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Why is Ns a normal derivative?
Recall the divergence theorem:
−
Z
Ω
∆u dx = −
Z
Ω
div(∇u) dx = −
Z
∂Ω
∂u
∂ν dσ, ∀ u ∈ C
2
(Ω).
For (−∆)s we have the following:
Z
Ω
(−∆)s
u dx = −
Z
RN \Ω
Nsu dx, ∀ u ∈ C
2
0
(R
N
).
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 24 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Why is Ns a normal derivative?
Green Formula: ∀ u ∈ C
2
(Ω) and ∀ v ∈ C
1
(Ω),
Z
Ω
∇u · ∇v dx = −
Z
Ω
v∆u dx +
Z
∂Ω
v
∂u
∂ν dσ.
For (−∆)s we have the following: ∀ u ∈ C
2
0
(R
N ) and v ∈ C
1
0
(R
N ),
CN,s
2
Z
R2N \(RN \Ω)2
(u(x) − u(y))(v(x) − v(y))
|x − y|
N+2s
dxdy
=
Z
Ω
v(−∆)s
u dx +
Z
RN \Ω
vNsu dx.
R
2N
\ (R
N
\ Ω)2 = (Ω × Ω) ∪ (Ω × (R
N
\ Ω)) ∪ ((R
N
\ Ω) × Ω).
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 25 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Why is Ns a normal derivative?
For every u, v ∈ C
2
0
(R
N ) we have that (Ros-Oton and Valdinoci, JEMS
2017)
lim
s↑1−
Z
RN \Ω
vNsu dx =
Z
∂Ω
v
∂u
∂ν dσ.
Observation
We have shown that Ns plays the same role for (−∆)s
that the classical
normal derivative ∂
∂ν does for ∆.
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 26 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
The Neumann problem for (−∆)s
1 f ∈ L
2
(Ω), g ∈ L
2
(∂Ω). The Neumann problem for ∆ is given by
−∆u = f in Ω,
∂u
∂ν = g on ∂Ω.
It is well-known that the above problem is well-posed if and only if
Z
Ω
f dx +
Z
∂Ω
g dσ = 0.
2 Let f ∈ L
2
(Ω) and g ∈ L
1
(R
N \ Ω). We consider the problem
(−∆)s
u = f in Ω, Nsu = g in R
N
\ Ω. (2.7)
What is a weak solution of (2.7)? When is (2.7) well-posed?
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs
Page 27 of 58
Objectives of the talk
Space-time fractional order operators
Controllability results for space-time fractional PDEs
The case of the fractional heat equation
Open problems
Another fractional order Sobolev space
Let g ∈ L
1
(R
N \ Ω) be fixed and let
W
s,2
Ω
:= n
u ∈L
2
(Ω), |g|
1
2 u ∈ L
2
(R
N
\ Ω),
Z Z
R2N \(RN \Ω)2
|u(x) − u(y)|
2
|x − y|
N+2s
dxdy < ∞
o
be endowed with the norm
kuk
2
Ws,2
Ω
:= Z
Ω
|u|
2
dx +
Z
RN \Ω
|g||u|
2
dx
+
Z Z
R2N \(RN \Ω)2
|u(x) − u(y)|
2
|x − y|
N+2s
dxdy.
Then W
s,2
Ω
is a Hilbert space (Ros-Oton and Valdinoci, JEMS 2017).
Mahamadi Jacob WARMAGeorge Mason University, Fairfax, Virginia (USA)The research of the author is partially supported by US Air Force Office of Scientific Research (AFOSR) US Army Research Office (ARO) Controllability of Fractional PDEs